Properties

Label 2-72e2-1.1-c1-0-87
Degree $2$
Conductor $5184$
Sign $-1$
Analytic cond. $41.3944$
Root an. cond. $6.43385$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5-s + 4·11-s + 5·13-s − 5·17-s − 8·19-s − 4·23-s − 4·25-s − 3·29-s − 4·31-s − 3·37-s − 6·41-s − 4·43-s − 12·47-s − 7·49-s + 10·53-s + 4·55-s − 8·59-s + 5·61-s + 5·65-s − 8·67-s + 16·71-s − 5·73-s + 4·79-s − 4·83-s − 5·85-s + 3·89-s − 8·95-s + ⋯
L(s)  = 1  + 0.447·5-s + 1.20·11-s + 1.38·13-s − 1.21·17-s − 1.83·19-s − 0.834·23-s − 4/5·25-s − 0.557·29-s − 0.718·31-s − 0.493·37-s − 0.937·41-s − 0.609·43-s − 1.75·47-s − 49-s + 1.37·53-s + 0.539·55-s − 1.04·59-s + 0.640·61-s + 0.620·65-s − 0.977·67-s + 1.89·71-s − 0.585·73-s + 0.450·79-s − 0.439·83-s − 0.542·85-s + 0.317·89-s − 0.820·95-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5184\)    =    \(2^{6} \cdot 3^{4}\)
Sign: $-1$
Analytic conductor: \(41.3944\)
Root analytic conductor: \(6.43385\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 5184,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 - T + p T^{2} \)
7 \( 1 + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 - 5 T + p T^{2} \)
17 \( 1 + 5 T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 3 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 3 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 - 5 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 - 16 T + p T^{2} \)
73 \( 1 + 5 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 3 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.039916880460102370529112408454, −6.86302109026996353454390617815, −6.37328189205345286122055890684, −5.96384835718157756460596018465, −4.84387611280887247576956835775, −3.98734145203664581686219175335, −3.54544618313642903226831863657, −2.05606192530423127635630654035, −1.63685953812129574992102885846, 0, 1.63685953812129574992102885846, 2.05606192530423127635630654035, 3.54544618313642903226831863657, 3.98734145203664581686219175335, 4.84387611280887247576956835775, 5.96384835718157756460596018465, 6.37328189205345286122055890684, 6.86302109026996353454390617815, 8.039916880460102370529112408454

Graph of the $Z$-function along the critical line